Place the beet in a saucepan; cover with cold unsalted water. Heat slowly to boiling; reduce the heat. Simmer, uncovered, until barely tender, about 35 minutes. Drain under cold water, Remove the skin, dice, and reserve.

Meanwhile, cook the potatoes, uncovered, in boiling salted water until barely tender, about 20 minutes. Drain; let stand 5 minutes. Peel and dice.

While the potatoes are still warm, combine them with the diced beet, the anchovy filets and oil, and the onion. Toss lightly.

To serve, divide the beet mixture into four portions and place each on a separate plate. With the back of a spoon, make a small indentation in each portion. Slide an egg yolk into each little "well" and sprinkle it with pepper. Each person mixes the egg yolk into the beets at the table before eating.